- List of equations in classical mechanics
- ;
**Nomenclature**:

**"a**" = acceleration (m/s²):**"g**" = gravitational field strength/acceleration in free-fall (m/s²):**"F**" = force (N = kg m/s²): "E"_{k}= kinetic energy (J = kg m²/s²): "E"_{p}= potential energy (J = kg m²/s²): "m" = mass (kg):**"p**" = momentum (kg m/s):**"s**" = displacement (m):**"R**" = radius (m): "t" = time (s):**"v**" = velocity (m/s):**"v**"_{0}= velocity at time t=0: "W" = work (J = kg m²/s²):**"τ**" = torque (m N, not J) (torque is the rotational form of force):**"s**"(t) = position at time t:**"s**"_{0}= position at time t=0:**"r**"_{unit}= unit vector pointing from the origin in polar coordinates:**"θ**"_{unit}= unit vector pointing in the direction of increasing values of theta in polar coordinatesNote: All quantities in bold represent vectors.

**Classical mechanics**is the branch ofphysics used to describe the motion ofmacroscopic objects. [*Harvnb|Mayer|Sussman|Wisdom|2001|p=xiii*] It is the most familiar of the theories of physics. The concepts it covers, such asmass ,acceleration , andforce , are commonly used and known. [*Harvnb|Berkshire|Kibble|2004|p=1*] The subject is based upon a three-dimensionalEuclidean space with fixed axes, called a frame of reference. The point of concurrency of the three axes is known as the origin of the particular space. [*Harvnb|Berkshire|Kibble|2004|p=2*]Classical mechanics utilises many

equation —as well as other mathematical concepts—which relate various physical quantities to one another. These includedifferential equations s,manifold s,Lie group s, andergodic theory . [*Harvnb|Arnold|1989|p=v*] This page gives a summary of the most important of these.**Equations****Velocity**: $mathbf\{v\}\_\{mbox\{average\; =\; \{Delta\; mathbf\{d\}\; over\; Delta\; t\}$: $mathbf\{v\}\; =\; \{dmathbf\{s\}\; over\; dt\}$

**Acceleration**: $mathbf\{a\}\_\{mbox\{average\; =\; frac\{Deltamathbf\{v\{Delta\; t\}$ : $mathbf\{a\}\; =\; frac\{dmathbf\{v\{dt\}\; =\; frac\{d^2mathbf\{s\{dt^2\}$

*Centripetal Acceleration

: $|mathbf\{a\}\_c\; |\; =\; omega^2\; R\; =\; v^2\; /\; R$("R" = radius of the circle, ω = "v/R"

angular velocity )Momentum : $mathbf\{p\}\; =\; mmathbf\{v\}$

**Force**:$sum\; mathbf\{F\}\; =\; frac\{dmathbf\{p\{dt\}\; =\; frac\{d(mmathbf\{v\})\}\{dt\}$

:$sum\; mathbf\{F\}\; =\; mmathbf\{a\}\; quad$ (Constant Mass)

**Impulse**:$mathbf\{J\}\; =\; Delta\; mathbf\{p\}\; =\; int\; mathbf\{F\}\; dt$:

$mathbf\{J\}\; =\; mathbf\{F\}\; Delta\; t\; quad$if**F**is constantMoment of inertia For a single

axis of rotation :The moment of inertia for an object is the sum of the products of the mass element and the square of their distances from the axis of rotation:$I\; =\; sum\; r\_i^2\; m\_i\; =int\_M\; r^2\; mathrm\{d\}\; m\; =\; iiint\_V\; r^2\; ho(x,y,z)\; mathrm\{d\}\; V$

Angular momentum :$|L|\; =\; mvr\; quad$ if

**v**is perpendicular to**r**Vector form:

:$mathbf\{L\}\; =\; mathbf\{r\}\; imes\; mathbf\{p\}\; =\; mathbf\{I\},\; omega$

(Note:

**I**can be treated like a vector if it is diagonalized first, but it is actually a 3×3 matrix - atensor of rank-2)**r**is the radius vector.Torque :$sum\; \backslash boldsymbol\{\; au\}\; =\; frac\{dmathbf\{L\{dt\}$:$sum\; \backslash boldsymbol\{\; au\}\; =\; mathbf\{r\}\; imes\; mathbf\{F\}\; quad$if |

**r**| and the sine of the angle between**r**and**p**remains constant.:$sum\; \backslash boldsymbol\{\; au\}\; =\; mathbf\{I\}\; \backslash boldsymbol\{alpha\}$This one is very limited, more added later.**α**= d**ω**/dt**Precession**Omega is called the precession angular speed, and is defined:

:$\backslash boldsymbol\{Omega\}\; =\; frac\{wr\}\{I\backslash boldsymbol\{omega$

(Note:

**w**is the weight of the spinning flywheel)**Energy**for "m" as a constant:

:$Delta\; E\_k\; =\; int\; mathbf\{F\}\_\{mbox\{net\; cdot\; dmathbf\{s\}\; =\; int\; mathbf\{v\}\; cdot\; dmathbf\{p\}\; =\; egin\{matrix\}frac\{1\}\{2\}end\{matrix\}\; mv^2\; -\; egin\{matrix\}frac\{1\}\{2\}end\{matrix\}\; m\{v\_0\}^2\; quad$

:$Delta\; E\_p\; =\; mgDelta\; h\; quad\; ,!$ in field of gravity

**Central force motion**: $frac\{d^2\}\{d\; heta^2\}left(frac\{1\}\{mathbf\{r\; ight)\; +\; frac\{1\}\{mathbf\{r\; =\; -frac\{mumathbf\{r\}^2\}\{mathbf\{l\}^2\}mathbf\{F\}(mathbf\{r\})$

Equations of motion (constant acceleration)These equations can be used only when acceleration is constant. If acceleration is not constant then

calculus must be used.:$v\; =\; v\_0+at\; ,$

:$s\; =\; frac\; \{1\}\; \{2\}(v\_0+v)\; t$

:$s\; =\; v\_0\; t\; +\; frac\; \{1\}\; \{2\}\; a\; t^2$

:$v^2\; =\; v\_0^2\; +\; 2\; a\; s\; ,$

:$s\; =\; vt\; -\; frac\; \{1\}\; \{2\}\; a\; t^2$

Derivation of these equation in vector format and without having is shown here These equations can be adapted for angular motion, where angular acceleration is constant:

: $omega\; \_1\; =\; omega\; \_0\; +\; alpha\; t\; ,$

: $heta\; =\; frac\{1\}\{2\}(omega\; \_0\; +\; omega\; \_1)t$

: $heta\; =\; omega\; \_0\; t\; +\; frac\{1\}\{2\}\; alpha\; t^2$

: $omega\; \_1^2\; =\; omega\; \_0^2\; +\; 2alpha\; heta$

: $heta\; =\; omega\; \_1\; t\; -\; frac\{1\}\{2\}\; alpha\; t^2$

**ee also***

Acoustics

*Classical mechanics

*Electromagnetism

*Mechanics

*Optics

*Thermodynamics **Notes****References***citation|title=Mathematical Methods of Classical Mechanics|last=Arnold|first=Vladimir I.|publisher=Springer|year=1989|isbn=978-0-387-96890-2|edition=2nd

*citation|title=Classical Mechanics|last1=Berkshire|last2=Kibble|first1=Frank H.|first2=T. W. B.|edition=5th|publisher=Imperial College Press|year=2004|isbn=978-1860944352

*citation|title=Structure and Interpretation of Classical Mechanics|last1=Mayer|last2=Sussman|last3=Wisdom|first1=Meinhard E.|first2=Gerard J.|first3=Jack|publisher=MIT Press|year=2001|isbn=978-0262194556**External links*** [

*http://www.astro.uvic.ca/~tatum/classmechs.html Lectures on classical mechanics*]

* [*http://scienceworld.wolfram.com/biography/Newton.html Biography of Isaac Newton, a key contributor to classical mechanics*]

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